Generalized Maximum Flows over Time

Flows over time and generalized flows are two advanced network flow models of utmost importance, as they incorporate two crucial features occurring in numerous real-life networks. Flows over time feature time as a problem dimension and allow to realistically model the fact that commodities (goods, information, etc.) are routed through a network over time. Generalized flows allow for gain/loss factors on the arcs that model physical transformations of a commodity due to leakage, evaporation, breeding, theft, or interest rates. Although the latter effects are usually time-bound, generalized flow models featuring a temporal dimension have never been studied in the literature. In this paper we introduce the problem of computing a generalized maximum flow over time in networks with both gain factors and transit times on the arcs. While generalized maximum flows and maximum flows over time can be computed efficiently, our combined problem turns out to be NP-hard and even completely non-approximable. A natural special case is given by lossy networks where the loss rate per time unit is identical on all arcs. For this case we present a (practically efficient) FPTAS.

[1]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[2]  K. Onaga.  Optimum flows in general communication networks , 1967 .

[3]  K. Onaga.  Dynamic Programming of Optimum Flows in Lossy Communication Nets , 1966 .

[4]  Sven Oliver Krumke,et al.  Generalized max flow in series-parallel graphs , 2013, Discret. Optim..

[5]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[6]  Tomasz Radzik.  Improving time bounds on maximum generalised flow computations by contracting the network , 2004, Theor. Comput. Sci..

[7]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[8]  Lisa Fleischer,et al.  Fast and simple approximation schemes for generalized flow , 2002, Math. Program..

[9]  K. Truemper.  On Max Flows with Gains and Pure Min-Cost Flows , 1977 .

[10]  Éva Tardos,et al.  Generalized maximum flow algorithms , 1999 .

[11]  Martin Skutella,et al.  Multicommodity flows over time: Efficient algorithms and complexity , 2007, Theor. Comput. Sci..

[12]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[13]  Éva Tardos,et al.  Efficient continuous-time dynamic network flow algorithms , 1998, Oper. Res. Lett..

[14]  Mark A. Fuller,et al.  Faster Algorithms for the Generalized Network Flow Problem , 1998 .

[15]  L. V. Kantorovich,et al.  Mathematical Methods of Organizing and Planning Production , 1960 .

[16]  Norman Zadeh,et al.  A bad network problem for the simplex method and other minimum cost flow algorithms , 1973, Math. Program..

[17]  Jay E. Aronson,et al.  A survey of dynamic network flows , 1989 .

[18]  Andrew V. Goldberg,et al.  Combinatorial Algorithms for the Generalized Circulation Problem , 1991, Math. Oper. Res..

[19]  Donald Goldfarb,et al.  A Faster Combinatorial Algorithm for the Generalized Circulation Problem , 1996, Math. Oper. Res..

[20]  Edward Minieka,et al.  Maximal, Lexicographic, and Dynamic Network Flows , 1973, Oper. Res..

[21]  Tomasz Radzik,et al.  Faster algorithms for the generalized network flow problem , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[22]  Martin Skutella,et al.  An Introduction to Network Flows over Time , 2008, Bonn Workshop of Combinatorial Optimization.

[23]  Martin Skutella,et al.  Minimum cost flows over time without intermediate storage , 2003, SODA '03.

[24]  David Gale,et al.  Transient flows in networks. , 1959 .

[25]  James B. Orlin,et al.  Polynomial-Time Highest-Gain Augmenting Path Algorithms for the Generalized Circulation Problem , 1997, Math. Oper. Res..

[26]  L. Lovász,et al.  Research trends in combinatorial optimization: Bonn 2008 , 2009 .

[27]  Martin Skutella,et al.  Quickest Flows Over Time , 2007, SIAM J. Comput..

[28]  Warren B. Powell,et al.  Stochastic and dynamic networks and routing , 1995 .

[29]  Michel Minoux,et al.  Graphs and Algorithms , 1984 .

[30]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[31]  D. R. Fulkerson,et al.  Constructing Maximal Dynamic Flows from Static Flows , 1958 .

[32]  Éva Tardos,et al.  Polynomial time algorithms for some evacuation problems , 1994, SODA '94.

[33]  David P. Williamson,et al.  A simple GAP-canceling algorithm for the generalized maximum flow problem , 2006, SODA '06.

[34]  Robert E. Tarjan,et al.  A faster deterministic maximum flow algorithm , 1992, SODA '92.

[35]  Kevin D. Wayne.  A Polynomial Combinatorial Algorithm for Generalized Minimum Cost Flow , 2002, Math. Oper. Res..

[36]  W. L. Wilkinson,et al.  An Algorithm for Universal Maximal Dynamic Flows in a Network , 1971, Oper. Res..

[37]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[38]  Gerhard J. Woeginger,et al.  Minimum-cost dynamic flows: The series-parallel case , 2004, Networks.